Entanglement and conservation laws in many body systems

One cannot overestimate the importance of entanglement as a fundamental aspect of quantum mechanics. Recently it has also gained a central role in the study of many body systems in condensed matter and high energy physics. After reviewing this, I will pose the main question of our recent studies: How are symmetries, which give rise to conservation laws, manifested by entanglement measures? Similarly to the system Hamiltonian, a subsystem's reduced density matrix is composed of blocks characterized by symmetry quantum numbers, or charge sectors. I will present a geometric method for extracting the contribution of individual charge sectors to a subsystem’s entanglement measures (entropies and negativities) within the replica approach, via threading of appropriate conjugate Aharonov-Bohm fluxes through a multi-sheeted Riemann surface.

Specializing to the case of 1+1D conformal field theory, I will describe a general exact result for the entanglement characteristics, in the ground state as well as following a quench. I will apply it to a variety of systems, ranging from free and interacting fermions to spin and parafermion chains, and verify it against exact asymptotic results and numerics. For example, I will show that the total ground-state entanglement entropy, which scales as the logarithm of the subsystem size, is composed of square-root of log contributions of individual subsystem charge sectors for interacting fermion chains, or even subsystem-size-independent contributions when total spin conservation is also accounted for. I will proceed to nonequilibrium current-carrying steady states, and show that they exhibit very unusual extensive entanglement between far-away segments with peculiar distribution between symmetry sectors.

I will conclude by describing how measurements of the contributions to the entanglement or negativity from separate charge sectors can be performed with ultracold bosons or fermions and similar systems, and how they could be calculated efficiently from tensor network states.